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Properties

Label 1-1183-1183.263-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.109 - 0.993i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0402 − 0.999i)2-s + (−0.970 − 0.239i)3-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (−0.200 + 0.979i)6-s + (0.120 + 0.992i)8-s + (0.885 + 0.464i)9-s + (0.120 − 0.992i)10-s + (0.885 − 0.464i)11-s + (0.987 + 0.160i)12-s + (−0.919 − 0.391i)15-s + (0.987 − 0.160i)16-s + (−0.919 − 0.391i)17-s + (0.428 − 0.903i)18-s + 19-s + (−0.996 − 0.0804i)20-s + ⋯
L(s)  = 1  + (−0.0402 − 0.999i)2-s + (−0.970 − 0.239i)3-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (−0.200 + 0.979i)6-s + (0.120 + 0.992i)8-s + (0.885 + 0.464i)9-s + (0.120 − 0.992i)10-s + (0.885 − 0.464i)11-s + (0.987 + 0.160i)12-s + (−0.919 − 0.391i)15-s + (0.987 − 0.160i)16-s + (−0.919 − 0.391i)17-s + (0.428 − 0.903i)18-s + 19-s + (−0.996 − 0.0804i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.109 - 0.993i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.109 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9096637536 - 0.8146240979i\)
\(L(\frac12)\) \(\approx\) \(0.9096637536 - 0.8146240979i\)
\(L(1)\) \(\approx\) \(0.7885733151 - 0.4581847538i\)
\(L(1)\) \(\approx\) \(0.7885733151 - 0.4581847538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.0402 - 0.999i)T \)
3 \( 1 + (-0.970 - 0.239i)T \)
5 \( 1 + (0.987 + 0.160i)T \)
11 \( 1 + (0.885 - 0.464i)T \)
17 \( 1 + (-0.919 - 0.391i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.845 + 0.534i)T \)
31 \( 1 + (-0.200 + 0.979i)T \)
37 \( 1 + (-0.200 + 0.979i)T \)
41 \( 1 + (0.692 - 0.721i)T \)
43 \( 1 + (0.948 + 0.316i)T \)
47 \( 1 + (0.428 + 0.903i)T \)
53 \( 1 + (0.799 - 0.600i)T \)
59 \( 1 + (0.987 + 0.160i)T \)
61 \( 1 + (0.120 - 0.992i)T \)
67 \( 1 + (0.568 - 0.822i)T \)
71 \( 1 + (0.278 + 0.960i)T \)
73 \( 1 + (-0.845 - 0.534i)T \)
79 \( 1 + (0.428 + 0.903i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.987 - 0.160i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.807917170990325841562462508517, −20.86901501842005846813312370599, −19.81454901578799546379414008505, −18.70816038162359259598172851658, −17.81152318287144372758615164611, −17.576546498107868322480682686419, −16.8292573120179841985719577385, −16.16580897038838746992760283505, −15.317755282353305308214635282693, −14.58018632133914775082475095420, −13.61394893429707113192405163437, −13.02275145659747483147472676562, −12.09841484066802818902071372363, −11.15734612744287673608776641761, −10.062635383019679975498233744622, −9.50653681975053893824562283706, −8.89303000560284444136639103608, −7.474730131213433035496055897, −6.8615939335222564371331573199, −5.79779427017528029920626609318, −5.67105264219988800341754524995, −4.45921896907264364889956823412, −3.8442467159889073019394997981, −1.98002748354871748160219238269, −0.88789108110360893086521261274, 0.830917114658708659125333079728, 1.67411370598185141186809175411, 2.60749898376575735058405017572, 3.79024518445740832929649037657, 4.80426360722020317471928127244, 5.55153414681795712313402769712, 6.38068943315635979303569086700, 7.266736220898946028926980441625, 8.64792661027376975581148838637, 9.38754422681368082530196260499, 10.14321348018336513495825748933, 10.95216726807884779515303038568, 11.48808245674648932650142520327, 12.39317861470774420011199310279, 13.03344736615326792346370411556, 13.92162449008729685617028115357, 14.374370373303330286129378695911, 15.86679151828263888848498644883, 16.754619547899513627245908641913, 17.408541351989197645342855706062, 18.07600364351060299967019027889, 18.55796614106620539224332577316, 19.47648726314811984930084104963, 20.36272379391398963402053312464, 21.11297305864645222031555641431

Graph of the $Z$-function along the critical line